MATH 6150F — Coxeter systems and Iwahori-Hecke algebras (Spring 2017) Lecture 10

1 Summary of facts so far

By now we have developed quite a number of general, useful properties of Coxeter systems.

Let (W,S) be a Coxeter system. Write m(s, t) for the order of st in W for s, t ∈ S, so that

W = 〈s ∈ S : (st)m(s,t) = 1 for s, t ∈ S with m(s, t) <∞〉.

Define V = R-span{αs : s ∈ S} as the real vector space with a basis indexed by S, and define (·, ·) as thebilinear form on V with (αs, αt) = − cos(π/m(s, t)) for s, t ∈ S.

For any α ∈ V , there is a unique expansion

α =∑s∈S

csαs

for some real coefficients cs ∈ R. Write α > 0 if α 6= 0 and every cs ≥ 0. Write α < 0 if −α > 0.

The root system of (W,S) is the set Φ = {wαs : w ∈W, s ∈ S} ⊂ V .

Let Φ+ = {α ∈ Φ : α > 0} and Φ− = {α ∈ Φ : α < 0}.

The following five theorems summarize most of our progress in the last few lectures.

Theorem (Geometric representation). The vector space V has a unique W -module structure in whichs ∈ S acts as the reflection s : v 7→ v − 2(αs, v)αs for v ∈ V. With respect to this structure, it holds that(wu,wv) = (u, v) for all w ∈W and u, v ∈ V . If w ∈W is such that wv = v for all v ∈ V then w = 1.

Theorem (Parabolic subgroups). Let J ⊂ S and define WJ = 〈s ∈ J〉 ⊂W

(i) (WJ , J) is a Coxeter system.

(ii) The length function of WJ (defined with respect to J) agrees with the length function of W .

(iii) The set S is a minimal generating set of W .

(iv) If w ∈WJ and w = s1 · · · sr (si ∈ S) is a reduced expression, then every si ∈ J .

Theorem (Geometric interpretation of length function). The following properties hold:

(a) If w ∈W and s ∈ S then `(ws) < `(w) if and only if wαs < 0.

(b) There is a disjoint union Φ = Φ+ t Φ−.

(c) If s ∈ S then sαs = −αs and sβ ∈ Φ+ for all α 6= β ∈ Φ+.

(d) The length `(w) of w ∈W is the number of root α ∈ Φ+ with wα ∈ Φ−.

Theorem (Reflections). The following properties hold:

(a) If wαs = w′αs′ = α ∈ Φ for some w,w′ ∈W and s, s′ ∈ S, then wsw−1 = w′s′(w′)−1.

Denote this element by sα ∈W , and let T = {sα : α ∈ Φ} = {wsw−1 : w ∈W, s ∈ S}.

(b) The correspondence α 7→ sα is a bijection Φ+ → T .

(c) If α, β ∈ Φ and β = wα for some w ∈W then wsαw−1 = sβ .

(d) If w ∈W and α ∈ Φ+ then `(wsα) > `(w) if and only if wα > 0.

Theorem (Strong exchange condition). Let w = s1 · · · sr (si ∈ S) with `(w) ≤ r. Suppose t ∈ T is suchthat `(wt) < `(w). Then there exists an index i ∈ [r] such that wt = s1 · · · si · · · sr. If `(w) = r, then theindex i is unique.

1

MATH 6150F — Coxeter systems and Iwahori-Hecke algebras (Spring 2017) Lecture 10

2 Bruhat order

This will be the most useful partial order on W which is compatible with ` : W → N.

Definition. The Bruhat order on W is partial order < which is the transitive closure of the relationwith w < wt whenever w ∈W and t ∈ T and `(wt) < `(w).

This means that if u, v ∈ W then u < v if and only if there are reflections t1, t2, . . . , tk ∈ T withv = ut1t2 · · · tk and `(u) < `(ut1) < `(ut1t2) < · · · < (ut1t2 · · · tk) = `(v).

Remark. Suppose <2 is the transitive closure of relation on W with w <2 tw for w ∈W and t ∈ T and`(w) < `(tw). Then < = <2 since

w <2 tw ⇔ w < wt′ for t′ = w−1tw ∈ T.

Thus despite appearances the definition of the Bruhat order is symmetric between left and right.

The left/right weak order on W is the partial order defined in the same way but requiring t ∈ S ⊂ T .This order is sometimes useful, but we won’t spend much time discussing it.

Note, if w < wt for w ∈ W and t ∈ T then `(wt) > `(w) but it is not required that `(wt) = `(w) + 1.Also, observe that if s ∈ S then w < ws if and only if `(w) < `(ws).

Example. If |S| = 2 so that W is a dihedral group and u, v ∈W , then u ≤ v if and only if `(u) ≤ `(v).(Try to work this our for yourself!)

Example. Let W = Sn and S = {(i, i+ 1) : i = 1, 2, . . . , n− 1} so that T = {(i, j) : 1 ≤ i < j ≤ n}. Wehave `(w(i, j)) > `(w) if and only if w(i) > w(j) for i < j. Therefore u < v in the Bruhat order of Sn ifand only if the sequence

v(1)v(2) · · · v(n)

can be obtained fromu(1)u(2) · · ·u(n)

by a sequence of moves in which we switch numbers in positions i and j (with i < j) with the ith numbergreater than the jth number.

For example, 24153→ 42153→ 45123→ 54123 so 24153 < 54123.

This doesn’t suggest a very efficient algorithm for checking whether u < v for u, v ∈ Sn. The followingresult of Deodhar, which we quote without proof, provides such an algorithm.

Proposition (Deodhar). If (a1, . . . , ak) is a sequence of integers, then write [a1, . . . , ak] for the sequencerewritten in increasing order. Define (a1, . . . , ak) � (b1, . . . , bk) if ai ≤ bi for all i.

Then u ≤ v in Sn if and only if [u(1), . . . , u(k)] � [v(1), . . . , v(k)] for 1 ≤ k ≤ n.

Let (W,S) be an arbitrary Coxeter system.

Proposition. Let u ≤ v and s ∈ S. Then us ≤ v or us ≤ vs (or both).

This turns out to be a key technical property, which is sometimes called the lifting property.

Proof. If suffices to assume that v = ut with t ∈ T and `(v) > `(u). If s = t then there is nothing toprove since then v = us. Assume s 6= t. There are two cases to consider:

(a) If `(us) = `(u)− 1 then us < u < v so us ≤ v.

(b) Suppose `(us) = `(u) + 1. Then ust′ = vs for t′ = sts, so it suffices to show that `(us) < `(vs),which will imply that us ≤ vs.

2

MATH 6150F — Coxeter systems and Iwahori-Hecke algebras (Spring 2017) Lecture 10

We argue by contradiction, so suppose `(vs) < `(us). We now apply the strong exchange conditionas follows. Note that for any reduced expression u = s1 · · · sr the expression us = s1 · · · srs is alsoreduced as `(us) > `(u).

Then vs = ust′ has a reduce expression given by omitting one factor from

s1 · srs.

The omitted factor cannot be s since s 6= t and v = ut, so we must have vs = s1 · · · si · · · srs forsome i so v = s1 · · · si · · · sr contradicting our assumption that `(v) > `(u).

This completes the proof of the proposition.

A subexpression of a reduced expression w = s1 · · · sr (si ∈ S) is any product of the form si1si2 · · · siqwhere 1 ≤ i1 < i2 < · · · < iq ≤ r.

Theorem. Let w = s1 · · · sr be a fixed but arbitrary reduced expression for w ∈ W . Then v ≤ w inBruhat order if and only if v can be obtained as a subexpression of the chosen reduced expression for w.

Proof. If w = vt for some t ∈ T , so that `(v) < `(w), then `(wt) < `(w) so by the strong exchangecondition v = s1 · · · si · · · sr for some i. Since the strong exchange condition does not require reducedexpressions, iterating the previous observation implies that if v < w then v is a subexpression of s1 · · · sr.

For the other direction, suppose v = si1 · · · viq where 1 ≤ i1 < i2 < · · · < iq ≤ r. We need to show thatv ≤ w. For this, proceed by induction on r = `(w). If r = 0 then v = w = 1 so v ≤ w.

Assume r > 0 and iq < r. Then, by the inductive hypothesis applied to s1 · · · sr−1, we get that

si1 · · · siq ≤ s1 · · · sr−1 < s1 · · · sr = w.

If r > 0 and iq = r, we first deduce by induction that

si1 · · · siq−1≤ s1 · · · sr−1

and then use the lifting property with s = siq = sr to conclude that

si1 · · · siq−1siq ≤ s1 · · · sr−1 <= w or si1 · · · siq−1siq ≤ s1 · · · sr−1sr = w.

In either case we get v ≤ w as desired.

The characterization of the Bruhat order in the theorem is a much more instructive way of thinking aboutthis order. However, it is awkward as an initial definition because of the apparent dependence on thechoice of reduced expression: this makes showing that < is transitive nontrivial.

As an application, we can answer one natural question about the Bruhat order of parabolic subgroups.

Corollary. If J ⊂ S then the Bruhat order of (WJ , J) agrees with the Bruhat order of (W,S) restrictedto WJ .

Proof. If w ∈WJ then w has a reduced expression in W involving only factors in J , and by the theoremv ≤ w in the Bruhat order of W or WJ if and only if v occurs as a subexpression of this reducedexpression.

Next time: a few more properties of the Bruhat order, and a description of a fundamental domain for Wacting on V .

3